Möbius function

There are three ways in which they are often mentioned

• classical Möbius function
• $\mu (n)=\{\begin{array}{lc}(-1{)}^k& (productofkdistinctprimeswherenisk)\\ 0& (otherwize)\end{array}$
• When n=1, it is the “product of 0 distinct primes”, so it is 1
• When n=4, it is not a “product of different prime numbers”, so it is 0

The “bias relation” ($a\prec b,a|b$) in which a divides b is semi-sequential (e.g. equations). The abstraction that we focus on here is adjacent algebra. The adjoint algebra zeta function is a function such that ζ(a,b) = 1 for all nonempty intervals $[a,b$], and the Mobius function is the multiplicative inverse of the zeta function.

With a concrete example of this adjacency algebra,

• 1: A whole positive integer with integer-divisibility relations
  • → attributed to classical Möbius function
• 2: The set (power set) formed by the entire subset of a finite set with inclusion relations
  • → Möbius function defined by the difference in set size
    • principle of inclusion leads to
• 3: Entering a greater-than or less-than relationship for the whole set of natural numbers
  • →0,0,…,0,-1,1
  • The zeta function corresponds to the cumulative sum, and the Möbius function corresponds to its inverse. difference operator.

https://ja.wikipedia.org/wiki/隣接代数_(order-theory)

fast zeta transformation in Sum over Subsets is 2, and the approximate version of the fast zeta transformation O(N log(log(N))) - noshi91’s note https://noshi91.hatenablog.com/entry/2018/12/ 27/121649] is 1

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